The people who love and care about you are concerned because they can see the pattern of your behavior that you may not be able to see right now.
What I can tell you is that if a young 15 or 16 year old young woman came into my office for a consultation and told me what you so bravely explained here, I would know three things: The first is that there is a pattern of behavior that orbits around gaining weight; secondly that the exercise burns what was consumed, which means there are calories being lost every day. (The normal processes of living and breathing and walking and talking, etc. use calories and if daily activities and exercise burn what was consumed there will be a systematic ongoing weight loss.) Finally a person who purges, keeps it a secret and thinks it is okay shows a contradiction in their behavior. If it really is okay—why not tell those who love you? Why keep the secret?
These things would prompt me to ask the young woman to talk to the psychiatrist and parents about the purging and to get a nutritional consult.
I believe these actions would be the right course of action for you to take at this time. By the way, I notice that you close your letter by saying that if there is nothing wrong you would show your parents and psychiatrist. I hope you can you show them what I have written here.